| L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.0763 − 0.873i)3-s + (−0.173 + 0.984i)4-s + (2.23 + 0.00416i)5-s + (0.619 − 0.619i)6-s + (−0.122 − 0.0573i)7-s + (−0.866 + 0.500i)8-s + (2.19 − 0.387i)9-s + (1.43 + 1.71i)10-s + (3.47 − 2.00i)11-s + (0.873 + 0.0763i)12-s + (−6.64 − 1.17i)13-s + (−0.0351 − 0.131i)14-s + (−0.167 − 1.95i)15-s + (−0.939 − 0.342i)16-s + (0.834 + 4.73i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.0441 − 0.504i)3-s + (−0.0868 + 0.492i)4-s + (0.999 + 0.00186i)5-s + (0.253 − 0.253i)6-s + (−0.0464 − 0.0216i)7-s + (−0.306 + 0.176i)8-s + (0.732 − 0.129i)9-s + (0.453 + 0.542i)10-s + (1.04 − 0.604i)11-s + (0.252 + 0.0220i)12-s + (−1.84 − 0.324i)13-s + (−0.00938 − 0.0350i)14-s + (−0.0431 − 0.504i)15-s + (−0.234 − 0.0855i)16-s + (0.202 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95722 + 0.317558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95722 + 0.317558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 5 | \( 1 + (-2.23 - 0.00416i)T \) |
| 37 | \( 1 + (4.92 - 3.57i)T \) |
| good | 3 | \( 1 + (0.0763 + 0.873i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (0.122 + 0.0573i)T + (4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 2.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.64 + 1.17i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.834 - 4.73i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.109 + 1.25i)T + (-18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.73 + 0.733i)T + (25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-1.83 - 1.83i)T + 31iT^{2} \) |
| 41 | \( 1 + (7.25 + 1.27i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 9.15iT - 43T^{2} \) |
| 47 | \( 1 + (0.530 - 0.142i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (9.89 - 4.61i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (1.04 + 2.24i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (4.63 - 3.24i)T + (20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (6.11 - 13.1i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (3.92 + 3.29i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.72 - 2.72i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.893 + 0.416i)T + (50.7 + 60.5i)T^{2} \) |
| 83 | \( 1 + (11.7 + 8.21i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.77 + 2.69i)T + (57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (6.86 - 11.8i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.75108897982444835777261305220, −10.39452717758951260388772444026, −9.560598123439304911923882679124, −8.635456595100912870590889557382, −7.28259673937338917838069900951, −6.75495111758768868023725460834, −5.70857698160358280419563629137, −4.73602592778534061341413020731, −3.24024419087627343407311899704, −1.64232751809468113300431651224,
1.70447400107214111935988832116, 2.98364258950205892332606540686, 4.65096056752504769291416418414, 4.97993085870845648501932146819, 6.54179865298585048444084407609, 7.29138213743720143952884261905, 9.267451933467723034657476281986, 9.553696420015631225926449598062, 10.30009355263104801108151108757, 11.34935342546585291934114686268
